i think this is a deeper question than most people realize, and quite crucial to the foundation of mathematics. perhaps one day, existence of the reals will be one possible mathematics, just as allowing or denying the axiom of choice gives rise to different fundamental theories of sets.

if at some level of existence, space/time/”stuff” no longer dispays the properties of continuousness we take for granted on the scale(s) we habitually inhabit, and are no longer valid, we may want to reconsider our notions of how we model space and time.

i dispute the validity of the statement pi is computable. as i understand it, a program written to calculate pi will NEVER terminate. just saying you can get the error of such a calculation as small as you like, may not be good enough. one of the troubles with real numbers is that the vast majority of them aren’t even possible to name. in fact, given any 2 rational numbers, no matter how close together, you cannot even name the real numbers between them.

now, i’m not advocating doing away with all the useful results the real number system (and the calculi it has spawned) has given us. i’m saying that the way(s) we have come up with the real number system are indicative of a lack of genuine understanding into the nature of the world we live in. i fear that we will come to regard them as a problem solved, rather than a mystery to continue to be explored.